Effect of Lankford Coefficients on Springback Behavior during Deep Drawing of Stainless Steel Cylinders

Accurate prediction of springback is increasingly required during deep-drawing formation of anisotropic stainless steel sheets. The anisotropy of sheet thickness direction is very important for predicting the springback and final shape of a workpiece. The effect of Lankford coefficients (r00, r45, r90) with different angles on springback was investigated using numerical simulation and experiments. The results show that the Lankford coefficients with different angles each have a different influence on springback. The diameter of the straight wall of the cylinder along the 45-degree direction decreased after springback, and showed a concave valley shape. The Lankford coefficient r90 had the greatest effect on the bottom ground springback, followed by r45 and then r00. A correlation was established between the springback of workpiece and Lankford coefficients. The experimental springback values were obtained by using a coordinate-measuring machine and showed good agreement with the numerical simulation results.


Introduction
As an important metal-forming process, sheet metal stamping is widely applied in the modern industry [1,2]. Springback is an inevitable physical phenomenon during the metal sheet-forming process [3][4][5]. The influence of springback on the accuracy and tolerance of a dimension is remarkable. The traditional trial-and-error and empirical methods for weakening springback and obtaining height-precision parts are time-consuming and expensive. The occurrence of defects, such as wrinkling, cracking, and springback, during sheet formation can be predicted with numerical simulations [6][7][8][9]. However, the predictions of springback and the final shape of the workpieces have a low accuracy rate because of the strong plastic anisotropy in the thickness direction.
A lot of research has been carried out in order to understand the influence of material properties and process parameters on springback behavior. Huang [10] analyzed the effects of different process parameters on springback during the stamping process using finite element numerical simulations. Minh [11] also used finite element simulation to analyze the effects of various factors-such as the blank holder force, friction coefficient, and blank thickness-on the springback of high-strength steel. Based on the numerical simulation results, it was evident that the blank holder force and blank thickness were the main factors affecting springback. Hashem and Roohi [12] utilized a numerical simulation to determine the effect of die and punch profile radii, as well as blank holder force on the springback and thinning percentage in the deep-drawing process of the cylindrical parts. The results show that an increased springback is observed due to an increased punch radius, and punch corner radius has been identified as the most significant effect on springback. Lajarin [13] found the blank holder force to be the most influential parameter for the springback of high-strength steel, followed by the die radius and friction conditions. Starman [14] formation using the finite element numerical simulation method and experimental methods, the research on springback of anisotropic sheet metal is still mostly confined to the forming of V/U-shaped parts, and there is little research on other common forming parts, for example, cylindrical cups. The influence of Lankford coefficients with different angles on springback during the cylinder deep-drawing process has not been clearly researched.
In this paper, a cylinder deep-drawing process with anisotropic stainless steel sheets was simulated based on the Barlat-Lian 1989 anisotropy yield criterion [26] by using Dynaform 5.9 software, and was used to predict springback. The Taguchi and ANOVA techniques were utilized to establish the correlation between springback at different angles from the rolling direction and Lankford coefficients (r 00 , r 45 , r 90 ) of 304 stainless steel. The ANOVA showed that the Lankford coefficient had a significant effect on springback. This research shows that each Lankford coefficient has an obvious influence on springback in diffident angles from the rolling direction by using the experimental and numerical simulation.

FEM Simulation Procedure
In this paper, the finite element numerical simulation was carried out on Dynaform. Dynaform software is a special piece of software jointly developed by ETA and LSTC for numerical simulation of sheet metal formation. It is a combination of LS-DYNA solver and ETA/FEMB front and back processor, and it is one of the most popular CAE tools for sheet metal formation and die design. Figure 1 shows the cylinder deep-drawing die; the actual object of the model is thecooking pot. The dimensions of the blank, die, punch, and blank holder are given in Table 1. One quarter of the 3D numerical model can be applied to the FEM model. The simulations require a large amount of computational time if they are not simplified, but they can provide a greater degree of precision. In this paper, a complete 3D numerical model was used. The 3D numerical model is shown in Figure 2. Table 2 shows the Lankford coefficients of metal sheets in different rolling directions with two horizontal factors set. The other mechanical properties of the materials were imported from the materials library in Dynaform. The punch and die were set as rigid, and the velocity of the punch was set at 2000 mm/s. The friction coefficient between the tools and the blank were set to 0.125. The contact-one-way surface-to-surface mode was employed to determine the friction type, and the adaptive meshing method was adopted to mesh the geometry model [27]. The full integrated planar shell was used, and the element type was defined as the time-efficient full-order integral Belytschko-Tsay shell element. This allowed for the adoption of four-point integration in order to avoid the appearance of "hourglassing" mode. A dynamic explicit algorithm was used to calculate the forming process. The implicit algorithm was applied to calculate the springback process. on springback during the cylinder deep-drawing process has not been clearly r In this paper, a cylinder deep-drawing process with anisotropic stainless s was simulated based on the Barlat-Lian 1989 anisotropy yield criterion [26] by u aform 5.9 software, and was used to predict springback. The Taguchi and AN niques were utilized to establish the correlation between springback at differ from the rolling direction and Lankford coefficients (r00, r45, r90) of 304 stainless ANOVA showed that the Lankford coefficient had a significant effect on spring research shows that each Lankford coefficient has an obvious influence on spr diffident angles from the rolling direction by using the experimental and num ulation.

FEM Simulation Procedure
In this paper, the finite element numerical simulation was carried out on D Dynaform software is a special piece of software jointly developed by ETA and numerical simulation of sheet metal formation. It is a combination of LS-DYNA ETA/FEMB front and back processor, and it is one of the most popular CAE too metal formation and die design. Figure 1 shows the cylinder deep-drawing die object of the model is thecooking pot. The dimensions of the blank, die, punch, holder are given in Table 1. One quarter of the 3D numerical model can be app FEM model. The simulations require a large amount of computational time if th simplified, but they can provide a greater degree of precision. In this paper, a 3D numerical model was used. The 3D numerical model is shown in Figure  shows the Lankford coefficients of metal sheets in different rolling directions horizontal factors set. The other mechanical properties of the materials were from the materials library in Dynaform. The punch and die were set as rigid, a locity of the punch was set at 2000 mm/s. The friction coefficient between the the blank were set to 0.125. The contact-one-way surface-to-surface mode was to determine the friction type, and the adaptive meshing method was adopte the geometry model [27]. The full integrated planar shell was used, and the ele was defined as the time-efficient full-order integral Belytschko-Tsay shell ele allowed for the adoption of four-point integration in order to avoid the app "hourglassing" mode. A dynamic explicit algorithm was used to calculate th process. The implicit algorithm was applied to calculate the springback proces     The material was modeled as an elastic-plastic material. The anisotropic characte tic was described by the Barlat-Lian 1989 anisotropic yield criterion [28]. The Barlat-L 1989 anisotropic yield criterion and the Hosford series' yield criterion were used to a lyze the plastic flow law of the drawing process [29][30][31]. Three stress-strain curves w obtained from the tensile test for the model material, as shown in Figure 3. The diffe curves were determined according to the ratio of the Lankford coefficients in each di tion of the actual material.   The material was modeled as an elastic-plastic material. The anisotropic characteristic was described by the Barlat-Lian 1989 anisotropic yield criterion [28]. The Barlat-Lian 1989 anisotropic yield criterion and the Hosford series' yield criterion were used to analyze the plastic flow law of the drawing process [29][30][31]. Three stress-strain curves were obtained from the tensile test for the model material, as shown in Figure 3. The different curves were determined according to the ratio of the Lankford coefficients in each direction of the actual material.  The material was modeled as an elastic-plastic material. The anisotropic characteristic was described by the Barlat-Lian 1989 anisotropic yield criterion [28]. The Barlat-Lian 1989 anisotropic yield criterion and the Hosford series' yield criterion were used to analyze the plastic flow law of the drawing process [29][30][31]. Three stress-strain curves were obtained from the tensile test for the model material, as shown in Figure 3. The different curves were determined according to the ratio of the Lankford coefficients in each direction of the actual material.

Taguchi Technique
The Taguchi technique was applied to the design scheme of the numerical simulation [32]. The two levels of the three-parameter orthogonal design, considering interactions (2 7 ), are presented in Table 3. The springback of different angles from the rolling direction was the process response. In order to understand the influence of Lankford coefficients, the ANOVA technique was applied to illustrate the degree of significance of each Lankford coefficient, including interactions.  Figure 4 shows the typical shape characteristics and measurement locations of the cylindrical cup. After formation, the workpiece was measured using CMM. Angles (α) were measured every 45 degrees from the rolling direction, and diameters were measured every 15 mm in the five sections along the height. A diagrammatic sketch of angles from the rolling direction is shown in Figure 4.

Taguchi Technique
The Taguchi technique was applied to the design scheme of the numerical simulation [32]. The two levels of the three-parameter orthogonal design, considering interactions (2 7 ), are presented in Table 3. The springback of different angles from the rolling direction was the process response. In order to understand the influence of Lankford coefficients, the ANOVA technique was applied to illustrate the degree of significance of each Lankford coefficient, including interactions.  Figure 4 shows the typical shape characteristics and measurement locations of the cylindrical cup. After formation, the workpiece was measured using CMM. Angles (α) were measured every 45 degrees from the rolling direction, and diameters were measured every 15 mm in the five sections along the height. A diagrammatic sketch of angles from the rolling direction is shown in Figure 4.

Formation Analysis
The forming limit diagram (FLD) and thickness change diagram can intuitively show the dynamic drawing process of the sheet metal and predict the formation of defects, such as cracking and wrinkling, and the thickness distribution of the sheet metal [33]. Figure 5a shows the forming limit diagram of the cylindrical cup after deep-drawing formation, Figure 5b shows the forming limit diagram of the cylindrical cup after springback, and Figure 5c is the cloud diagram of springback change in the cylindrical cup after springback calculation. It can be seen that the cylindrical cup fluctuates after springback with different degrees in the flange. The springback is apparent at 0 • , 45 • , and 90 • positions, which shows a cyclical trend of first decreasing and then increasing along the rolling direction. The straight wall of the cylinder also showed uneven springback.

Formation Analysis
The forming limit diagram (FLD) and thickness change diagram can intuitively show the dynamic drawing process of the sheet metal and predict the formation of defects, such as cracking and wrinkling, and the thickness distribution of the sheet metal [33]. Figure  5a shows the forming limit diagram of the cylindrical cup after deep-drawing formation, Figure 5b shows the forming limit diagram of the cylindrical cup after springback, and Figure 5c is the cloud diagram of springback change in the cylindrical cup after springback calculation. It can be seen that the cylindrical cup fluctuates after springback with different degrees in the flange. The springback is apparent at 0°, 45°, and 90° positions, which shows a cyclical trend of first decreasing and then increasing along the rolling direction. The straight wall of the cylinder also showed uneven springback.
(a) (c) (b) Due to the uneven springback deformation of the straight wall of the cylindrical cup, the sections with heights of 45 mm and 60 mm were selected for measurement, and 120 coordinate points were measured for each section. The difference between coordinate values of data points before and after springback was calculated. The cross-section difference point cloud diagrams are shown in Figure 6. The co-ordinates only represent the position of data points on the section of the cylindrical drawing section. The distance between each point and the origin represents the springback value. It can be seen that the springback difference between the two heights is similar, and is in the range of 0.150-25 mm. Within Due to the uneven springback deformation of the straight wall of the cylindrical cup, the sections with heights of 45 mm and 60 mm were selected for measurement, and 120 coordinate points were measured for each section. The difference between coordinate values of data points before and after springback was calculated. The cross-section difference point cloud diagrams are shown in Figure 6. The co-ordinates only represent the position of data points on the section of the cylindrical drawing section. The distance between each point and the origin represents the springback value. It can be seen that the springback difference between the two heights is similar, and is in the range of 0.150-25 mm. Within the angle of 0-45 • from the rolling direction, the springback difference firstly decreases, and then it increases. At the position of the maximum plastic strain value of r 45 , the angle of 0-45° from the rolling direction, the springback difference firstly decreases, and then it increases. At the position of the maximum plastic strain value of r45, that is, at the positions 45°, 135°, 225°, and 315° from the rolling direction, the springback difference of the cylinder drawing part reaches its maximum value.

Stress-Strain Analysis
The straight wall of the cylindrical cup is an area of force transmission during deep drawing, and no more plastic deformation occurs. The straight wall experiences a single axial tensile stress. There is a small amount of axial elongation and deformation. The state of stress and strain during deep drawing is shown in Figure 7. The first principal stress and strain of the model of the straight wall model's middle layer was extracted to analyze the reasons for uneven springback. The stress-strain analysis diagrams of cylindrical deep drawing at the heights of 45 mm and 60 mm are shown in Figure 8. The stress-strain data of 60 points on the circumference of the straight wall were extracted, and the red circle represents the average stressstrain value of all points. It can be seen that, at the height of 45 and 60 mm, the first principal stress was greater than the other two directions at the position of 45° from the rolling direction, while the first principal strain was smaller than the other two directions.
The stress-strain values for the three rolling directions of 0°, 45°, and 90° were compared and analyzed, and the results are shown in Table 4. The stress at 45° at the height

Stress-Strain Analysis
The straight wall of the cylindrical cup is an area of force transmission during deep drawing, and no more plastic deformation occurs. The straight wall experiences a single axial tensile stress. There is a small amount of axial elongation and deformation. The state of stress and strain during deep drawing is shown in Figure 7. The first principal stress and strain of the model of the straight wall model's middle layer was extracted to analyze the reasons for uneven springback.

Stress-Strain Analysis
The straight wall of the cylindrical cup is an area of force transmission drawing, and no more plastic deformation occurs. The straight wall experie axial tensile stress. There is a small amount of axial elongation and deformat of stress and strain during deep drawing is shown in Figure 7. The first pr and strain of the model of the straight wall model's middle layer was extract the reasons for uneven springback. The stress-strain analysis diagrams of cylindrical deep drawing at the mm and 60 mm are shown in Figure 8. The stress-strain data of 60 points on ference of the straight wall were extracted, and the red circle represents the av strain value of all points. It can be seen that, at the height of 45 and 60 mm, t cipal stress was greater than the other two directions at the position of 45° fro direction, while the first principal strain was smaller than the other two dire The stress-strain values for the three rolling directions of 0°, 45°, and 9 pared and analyzed, and the results are shown in Table 4. The stress at 45° The stress-strain analysis diagrams of cylindrical deep drawing at the heights of 45 mm and 60 mm are shown in Figure 8. The stress-strain data of 60 points on the circumference of the straight wall were extracted, and the red circle represents the average stress-strain value of all points. It can be seen that, at the height of 45 and 60 mm, the first principal stress was greater than the other two directions at the position of 45 • from the rolling direction, while the first principal strain was smaller than the other two directions.
The stress-strain values for the three rolling directions of 0 • , 45 • , and 90 • were compared and analyzed, and the results are shown in Table 4. The stress at 45 • at the height of 45 mm is 23% higher, and the stress at 45 • at the height of 60 mm is 19.37% higher than that of the other rolling directions. This is because the hardening curves are for different rolling directions. The value of the hardening curve at 45 • from the rolling direction was larger, and the stress value required during deep drawing was larger. The strain in the 45 • direction was smaller and contained more elastic stress in the deformation process, resulting in greater springback deformation after unloading. of 45 mm is 23% higher, and the stress at 45° at the height of 60 mm is 19.37% higher than that of the other rolling directions. This is because the hardening curves are for different rolling directions. The value of the hardening curve at 45° from the rolling direction was larger, and the stress value required during deep drawing was larger. The strain in the 45° direction was smaller and contained more elastic stress in the deformation process, resulting in greater springback deformation after unloading.

Boundary Inflow Analysis
The diagram of inflow of the cylindrical cup boundary material is shown in Figure 9. It shows a cyclical trend of first increasing and then decreasing between 0 • and 90 • from the rolling direction. At the positions 45 • , 135 • , 215 • , and 315 • , there was a larger inflow, and the maximum value was 40.76 mm. The Lankford coefficients in the 45 • direction were greater than those for the 0 • and 90 • directions. When the Lankford coefficients were large, the deformation resistance of the flange of the metal sheet was reduced, and the material flowed more easily. The flow stress value in the 45 • direction was large, so the inflow of material was larger. The flow stress in the 0 • and 90 • directions was smaller, elongation deformation was easier, and the inflow was smaller. This may be one of the reasons for the greater springback difference in this direction.

Boundary Inflow Analysis
The diagram of inflow of the cylindrical cup boundary material is shown in Figure 9. It shows a cyclical trend of first increasing and then decreasing between 0° and 90° from the rolling direction. At the positions 45°, 135°, 215°, and 315°, there was a larger inflow, and the maximum value was 40.76 mm. The Lankford coefficients in the 45° direction were greater than those for the 0° and 90° directions. When the Lankford coefficients were large, the deformation resistance of the flange of the metal sheet was reduced, and the material flowed more easily. The flow stress value in the 45° direction was large, so the inflow of material was larger. The flow stress in the 0° and 90° directions was smaller, elongation deformation was easier, and the inflow was smaller. This may be one of the reasons for the greater springback difference in this direction. The sheet firstly underwent elastic deformation, and then plastic deformation occurred after the stress exceeded the flow stress during the deformation process. After unloading, the internal stress was redistributed, and then springback occurred. The deformation and plastic deformation of the cylindrical cup drawing in the 45° direction was less severe than the other two directions. The circumferential stress in the 45° direction was relatively large, resulting in larger springback deformation at this position. The diameter of the cylinder after springback in the 45° direction was smaller, showing a greater springback difference.

Experimental Set-Up
Two different stainless steel sheets with the same thickness were selected for the experimental test. Based on the previous experimental tests [34,35], strong anisotropic properties were present in the two materials. Lankford coefficients of r00, r45, and r90 are listed in Table 2. For cylinder deep drawing, circular blanks with a diameter of 315 mm and a thickness of 0.6 mm were prepared. Figure 10 shows the drawing die designed and fabricated based on the simulation model. The sheet firstly underwent elastic deformation, and then plastic deformation occurred after the stress exceeded the flow stress during the deformation process. After unloading, the internal stress was redistributed, and then springback occurred. The deformation and plastic deformation of the cylindrical cup drawing in the 45 • direction was less severe than the other two directions. The circumferential stress in the 45 • direction was relatively large, resulting in larger springback deformation at this position. The diameter of the cylinder after springback in the 45 • direction was smaller, showing a greater springback difference.

Experimental Set-Up
Two different stainless steel sheets with the same thickness were selected for the experimental test. Based on the previous experimental tests [34,35], strong anisotropic properties were present in the two materials. Lankford coefficients of r 00 , r 45 , and r 90 are listed in Table 2. For cylinder deep drawing, circular blanks with a diameter of 315 mm and a thickness of 0.6 mm were prepared. Figure 10 shows the drawing die designed and fabricated based on the simulation model.

Experimental Results
After calculating the weighted average of the co-ordinates of two types of stainless steel workpieces at different heights, the radius values at different angles were obtained, as shown in Figure 11. The average values of the cross-section point cloud can be compared at the height of 30 mm, 45 mm, and 60 mm. The valley shape of the depression was clearer and more obvious in the material with the larger Lankford coefficients at the positions of 45°, 135°, 225°, and 315°. At the height of 15 mm near the bottom of the cylinder, the low-Lankford-coefficient material showed a more rounded cross-section. The high-Lankford-coefficient material showed a less rounded cross-section after drawing, and it was close to an ellipse along the long axis of the rolling direction and the short axis perpendicular to the rolling direction. At a height of 75 mm near the flange, the sections of both kinds of stainless steel showed an elliptical shape after deep drawing and springback. The experimental results are in good agreement with the FEM simulation results.

Experimental Results
After calculating the weighted average of the co-ordinates of two types of stainless steel workpieces at different heights, the radius values at different angles were obtained, as shown in Figure 11. The average values of the cross-section point cloud can be compared at the height of 30 mm, 45 mm, and 60 mm. The valley shape of the depression was clearer and more obvious in the material with the larger Lankford coefficients at the positions of 45 • , 135 • , 225 • , and 315 • . At the height of 15 mm near the bottom of the cylinder, the low-Lankford-coefficient material showed a more rounded cross-section. The high-Lankford-coefficient material showed a less rounded cross-section after drawing, and it was close to an ellipse along the long axis of the rolling direction and the short axis perpendicular to the rolling direction. At a height of 75 mm near the flange, the sections of both kinds of stainless steel showed an elliptical shape after deep drawing and springback. The experimental results are in good agreement with the FEM simulation results.

Experimental Results
After calculating the weighted average of the co-ordinates of two types of stainless steel workpieces at different heights, the radius values at different angles were obtained, as shown in Figure 11. The average values of the cross-section point cloud can be compared at the height of 30 mm, 45 mm, and 60 mm. The valley shape of the depression was clearer and more obvious in the material with the larger Lankford coefficients at the positions of 45°, 135°, 225°, and 315°. At the height of 15 mm near the bottom of the cylinder, the low-Lankford-coefficient material showed a more rounded cross-section. The high-Lankford-coefficient material showed a less rounded cross-section after drawing, and it was close to an ellipse along the long axis of the rolling direction and the short axis perpendicular to the rolling direction. At a height of 75 mm near the flange, the sections of both kinds of stainless steel showed an elliptical shape after deep drawing and springback. The experimental results are in good agreement with the FEM simulation results. Tables 5-8 compare the radius values along the three rolling directions at five section heights obtained from numerical simulation and experiments. It can be seen that the difference in radius between different rolling directions is more obvious in the simulation, and it was the largest at the heights of 45 mm and 60 mm. In the material with high-level Lankford coefficients, the differences reached 0.433 mm and 0.318 mm, respectively. In the material with low-level Lankford coefficients, the differences reached 0.387 mm and 0.32 mm, respectively. The experimental difference between radii in different rolling directions was close to the simulation result in the material with high-level Lankford coefficients. The difference at the heights of 30 mm and 45 mm reached 0.158 mm and 0.204 mm, respectively. The experimental radius difference between different rolling directions in material with low-level Lankford coefficients was small, reaching 0.127 mm and 0.08 mm at the heights of 45 mm and 60 mm, respectively.  Tables 5-8 compare the radius values along the three rolling directions at five section heights obtained from numerical simulation and experiments. It can be seen that the difference in radius between different rolling directions is more obvious in the simulation, and it was the largest at the heights of 45 mm and 60 mm. In the material with high-level Lankford coefficients, the differences reached 0.433 mm and 0.318 mm, respectively. In the material with low-level Lankford coefficients, the differences reached 0.387 mm and 0.32 mm, respectively. The experimental difference between radii in different rolling directions was close to the simulation result in the material with high-level Lankford coefficients. The difference at the heights of 30 mm and 45 mm reached 0.158 mm and 0.204 mm, respectively. The experimental radius difference between different rolling directions in material with low-level Lankford coefficients was small, reaching 0.127 mm and 0.08 mm at the heights of 45 mm and 60 mm, respectively.   Figure 12 shows experimental measurements of diameter at five sections along the height. They have a similar trend. The section at the height of 15 mm was close to the radius of the punch nose. The section at the height of 75 mm was similar to the radius of the die shoulder. The fillet radius had a great influence on the workpiece diameter. The cross-section of the cylinder after deep drawing showed an oval shape after springback. The diameters at the height of 30, 45, and 60 mm showed a similar trend. The results showed that the r 45 Lankford coefficient is the maximum value. In addition, as the Lankford coefficient increased, the diameter decreased.   Figure 12 shows experimental measurements of diameter at five sections along the height. They have a similar trend. The section at the height of 15 mm was close to the radius of the punch nose. The section at the height of 75 mm was similar to the radius of the die shoulder. The fillet radius had a great influence on the workpiece diameter. The cross-section of the cylinder after deep drawing showed an oval shape after springback. The diameters at the height of 30, 45, and 60 mm showed a similar trend. The results showed that the r45 Lankford coefficient is the maximum value. In addition, as the Lankford coefficient increased, the diameter decreased.  Table 9 shows the results of springback prediction by FEM simulation, which shows that the springback of every angle from the rolling direction is without symmetrical characteristics.  Table 9 shows the results of springback prediction by FEM simulation, which shows that the springback of every angle from the rolling direction is without symmetrical characteristics. To investigate the degree of significance of the Lankford coefficients, the ANOVA technique was used to analyze the springback. The mean overall value S/N S/N is expressed as Equation (1), where k is the number of simulations. The range of two levels (SR j ) is shown in Equation (2). The sum of squares owing to the variations of the overall mean (SS) and the mean of the Lankford coefficients with interactions (SS j ) are expressed as Equations (3) and (4), respectively. The percentage values (%p-Value j ) were calculated using Equation (5), which is generally applied when measuring the degree of significance of each Lankford coefficient [36].

Effects of Process Parameters on Springback
The results of the range analysis and variance analysis are shown in Table 10. It is revealed that the influence of Lankford coefficients on springback is different at different angles. The source of r 00 had a critical effect on the springback at θ 000 from the rolling direction, the r 45 is the key factor causing springback at θ 045 and θ 315 , and r 90 is key at θ 135 , θ 180 , and θ 225 . Furthermore, r 45 × r 00 has significant values for springback at θ 090 because of interactions with Lankford coefficients. Meanwhile, r 90 × r 45 is the key factor of springback at θ 270 . The measurement error also easily affected the range analyses, and so the ANOVA technique was used to analyze the springback. The ANOVA results shown in Table 9 correspond well with the range analysis results. Based on these analysis results, it has been found that the interactions of Lankford coefficients at different angles from the rolling direction have a clear effect on the springback. Figure 13a-h shows the sensitivity analysis of the effect of Lankford coefficients on springback. When the interactions of the Lankford coefficients were not considered, the amount of springback decreased with r 90 and r 45 at all angles, and the amount of springback decreased with increasing r 00 , except at the angle of θ 000 . When considering the interactions, the amount of springback increased with increasing r 90 × r 45 . When the r 45 and r 90 increased simultaneously, the interaction of r 90 × r 45 hindered springback and caused it to decrease. The amount of springback increased with increasing r 45 × r 00 , except at the angle of θ 180 . The amount of springback decreased with increasing r 90 × r 00 , except at the angles of θ 045 and θ 180 . The results show that the influences of the Lankford coefficient on springback at different angles are interrelated and interact with each other.  Figure 13a-h shows the sensitivity analysis of the effect of Lankford coefficients on springback. When the interactions of the Lankford coefficients were not considered, the amount of springback decreased with r90 and r45 at all angles, and the amount of springback decreased with increasing r00, except at the angle of θ000. When considering the interactions, the amount of springback increased with increasing r90 × r45. When the r45 and r90 increased simultaneously, the interaction of r90 × r45 hindered springback and caused it to decrease. The amount of springback increased with increasing r45 × r00, except at the angle of θ180. The amount of springback decreased with increasing r90 × r00, except at the angles of θ045 and θ180. The results show that the influences of the Lankford coefficient on springback at different angles are interrelated and interact with each other.

Comparing the FEM Simulation and Experimental Results
The experimental results of two kinds of 304 stainless steel were measured using CMM, as shown in Table 11. The No. 1 and No. 8 FEM simulation results are shown in Table 12. Figure 14 shows the comparison of an average bottom fillet of the FEM simulation and experimental results at different angles from the rolling direction. The material with high-level Lankford coefficients had a larger amount of springback at 0, 90, and 270 degrees from the rolling direction. The experimental results have good agreement with the FEM simulation results, with the bottom fillet showing a similar trend. The springback of the cylinder bottom fillet occurred along the rolling direction, and there was an increasing trend with every 45° decrease.

Comparing the FEM Simulation and Experimental Results
The experimental results of two kinds of 304 stainless steel were measured using CMM, as shown in Table 11. The No. 1 and No. 8 FEM simulation results are shown in Table 12. Figure 14 shows the comparison of an average bottom fillet of the FEM simulation and experimental results at different angles from the rolling direction. The material with high-level Lankford coefficients had a larger amount of springback at 0, 90, and 270 degrees from the rolling direction. The experimental results have good agreement with the FEM simulation results, with the bottom fillet showing a similar trend. The springback of the cylinder bottom fillet occurred along the rolling direction, and there was an increasing trend with every 45 • decrease.   The comparison between the bottom fillets of materials with high-and low-level Lankford coefficients is shown in Figure 15. The experimental results have good agreement with the FEM simulation results, showing a similar trend in springback. The amount of the bottom fillet decreased with an increase in the Lankford coefficient at all locations, except for 0, 90, and 270 degrees. The trend is more apparent in the FEM simulation results.

Conclusions
The influence of Lankford coefficients on stainless steel cylindrical cups was investigated using both experiments and numerical simulation. The conclusions are as follows: The simulation and experimental results show that the Lankford coefficients had an obvious effect on the cross-section diameter. The flow velocity of the blank was different   The comparison between the bottom fillets of materials with high-and low-level Lankford coefficients is shown in Figure 15. The experimental results have good agreement with the FEM simulation results, showing a similar trend in springback. The amount of the bottom fillet decreased with an increase in the Lankford coefficient at all locations, except for 0, 90, and 270 degrees. The trend is more apparent in the FEM simulation results.

Conclusions
The influence of Lankford coefficients on stainless steel cylindrical cups was investigated using both experiments and numerical simulation. The conclusions are as follows: The simulation and experimental results show that the Lankford coefficients had an obvious effect on the cross-section diameter. The flow velocity of the blank was different

Conclusions
The influence of Lankford coefficients on stainless steel cylindrical cups was investigated using both experiments and numerical simulation. The conclusions are as follows: The simulation and experimental results show that the Lankford coefficients had an obvious effect on the cross-section diameter. The flow velocity of the blank was different because of the anisotropy of the metal sheet, which makes the stress-strain values accumulate in different directions during the deep-drawing process, and finally causes a clear difference in springback. The simulated springback value for the straight wall was between 0.15 mm and 0.25 mm. The maximum springback value was at the position of 45 • from the rolling direction. Specifically, the diameters at different height sections were related to the Lankford coefficients at different angles from the rolling direction, which were characterized by a concave valley in the 45 degree direction of the straight wall. The radius difference between the 45 degree rolling direction and the other two directions at each section height was between 0.1 mm and 0.3 mm.
The ANOVA results illustrated the influence of Lankford coefficients on the springback of the bottom fillet. The Lankford coefficient has different levels of effects on springback depending on the angle from the rolling direction. The 90-degree angle had the greatest influence, followed by the 45-degree, with 0 degrees having the least influence. The experimental results showed a similar trend to the simulation results. In addition, the springback of the bottom fillet decreased with the increasing overall Lankford coefficients.
The combination of the FEM simulation, the ANOVA technique, and the experimental study of the cylinder deep-drawing process is an effective method for studying the influence of Lankford coefficients on springback and predicting the final shape with high precision.
In this paper, the study of the effect of Lankford coefficients on springback of a cylindrical cup during the deep-drawing process remained at the macroscopic stage, and further analysis on the microscopic aspects was not carried out. The mechanism by which metal anisotropy influences the springback of a cylinder needs to be explored further. Although some characteristic rules regarding springback and cylindrical cup properties during deep drawing were obtained in this study, the analysis of the specific degree of influence of Lankford coefficients on springback properties is still in the preliminary stage. Therefore, the quantitative analysis of the influence of Lankford coefficients on the springback of the cylindrical cup during deep drawing will remain the focus of future research. Based on the previous research on the cylindrical cup, the springback of large complex thin-walled parts in deep drawing will be further explored.

Data Availability Statement:
The data presented in this study are available on request from the corresponding author. The data are not publicly available due to privacy.